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\title{\huge \bf The Intersection of Two Cylinders \footnote{This file is from the 3D-XploreMath project. \hfil\break Please see http://vmm.math.uci.edu/3D-XplorMath/index.html}}
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\LARGE


This shows the space curve defined implicitly 
as the intersection of the two cylinders:

 \[ y^2 + z^2  =  f\!f \]

and 

 \[ (\cos(aa) x + \sin(aa)y)^2 + (z-cc)^2 = gg \]

These two cylinders are made visible by displaying a random set of dots on each of them.

In the default settings the two cylinders touch and the
default morph rotates one of them by changing $aa$. 

  We find it interesting to change the radius of the 
smaller cylinder while the cylinders keep touching:
morph $gg$ up to $f\!f$ while keeping dd=0, since we 
compute 
$ cc = \sqrt{f\!f} - \sqrt{gg} + dd$
At $gg = f\!f$ the intersection curve degenerates into
two ellipses (for each $aa$).

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